Ziphion
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Post by Ziphion on Jul 1, 2014 12:31:57 GMT -8
Hello! I've been looking closely at the Simple System Demo 36-card deck, and analyzing the checks and x's for each color. Here's what I found (positive is checks, negative is x's):
(btw, I tried creating an html table, but it completely broke my post. Sorry for the image.)
Here's a plot of that data:
Not including crits, green has 24 successes and 6 failures, yellow has 18 successes and 12 failures, red has 12 successes and 18 failures, and blue has 6 successes and 24 failures. Almost but not exactly symmetric, and so you get some strange results; for example, the probability of yellow succeeding against green is 56.9%, but the probability of red succeeding against yellow is 57.3%. On average, rotating the deck gives you about a 7% edge, but it fluctuates depending on the specific colors. Not catastrophic, but I think we can do better. Try this instead:
Much prettier! Here, the probabilities of yellow vs green, red vs yellow, or blue vs red are all the same: 56.94%. Red vs green, or blue vs yellow: 63.89%. Even better, every rotation of the deck corresponds to a 6.94% increase in probability of success.
I haven't included multiple flips here because I've got an error in my math that I can't locate (the probabilities get complicated, since you can get a -5, for example, by flipping a -3 and a -2, OR, by flipping a tie that includes a -3, and breaking it with a -2, OR, by flipping a tie that includes a -2, and breaking it with a -3), but I'll work on that some more and post again with more analysis.
Also - in the rules as written, if any of the cards you flip are critical success or failure, you get an automatic success/failure. This means 2x or 3x green has a higher chance of getting a critical success than a blue does, and 2x or 3x blue has a higher chance of getting critical failure than a green does. That doesn't really make sense. I think if you’re flipping multiple green, critical success cards should become three-check cards. If you’re flipping multiple blue, critical failure cards should become three-x cards.
So, what do you think about all that?
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Ziphion
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Post by Ziphion on Jul 1, 2014 12:45:05 GMT -8
You know what? My math is a little bit off in the info I posted here. To determine color-vs-color win probability, I made it so that ties always went 50-50. That's not correct when the colors are different. Instead of trying to get answers analytically like this, I think I'll just write a simulation of the resolution deck and see what the exact probabilities are. Updates to follow!
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Post by Dashing Inventor on Jul 1, 2014 12:56:46 GMT -8
Charts! This is excellent, thank you for putting this together. In all play testing so far that I've been present for, and from what I've been told from play sessions where I was not present, the current distribution returns expected results. However, I think implementing the changes you suggest above will only serve to improve upon that. I'll have to do some testing to see for sure. I've also made a pretty significant change to critical success and failures that makes incorporating them into multiple flips much more straightforward (more on that in the coming weeks).
Keep in mind that critical successes are treated as such only if they are the last card flipped in a multiple flip, otherwise you simply add their checks/strikes to the total as with any other card. Also, even when rolling 3 10-sided die you can still get a 3, easily beaten by a single six sided die - my point is that whenever you introduce random probability into a system, you can always end up with some 'rogue', or improbable, results. In part this contributes to tension/excitement. By design the Resolution deck doesn't completely eliminate this, but it does allow for a finer degree of probability control. Again, it appears using your suggested distribution above will help this nicely.
Thanks again and I love to see this kind of stuff so feel free to share any more ideas you have! I'm interested to see the updates you mentioned.
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Ziphion
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Post by Ziphion on Jul 1, 2014 14:02:15 GMT -8
Ah, okay. Criticals are just single successes/failures unless they're the last card flipped. That seems like a nicer solution than mine. I'm looking forward to seeing the change you mentioned! By the way, I thought I'd ask: is there a reason why you don't just add up all checks and strikes on a multi-flip, treating the strikes as negative? Like flipping a 3-strike and a 2-check translates to a 1-strike? Is it because you wanted to avoid zeroes?
I've been considering all results in my fiddlings with the numbers, including improbable ones, but because I've been trying to solve for all of the probabilities analytically, I made a couple of hard-to-find errors in my "multiple flip" spreadsheet. Once I make a properly functioning resolution deck simulator, I'll be able to get the correct numbers computationally and make all kinds of cool and useful plots. I'll also send you the file and source code for the simulator when it's ready (gimme a few days, maybe a week).
I love doing this kind of thing, by the way. I had toyed with making a 2d6 RPG system a couple of months ago before I'd heard of Simple System, and had a lot of fun simulating combat between two characters with different stats and abilities, and ironing out imbalances. My design goal was to make a straightforward system that was easy to pick up and play (as a reaction against the crazy complicated D&D 4e), but still had a strong foundation that was mathematically well-understood. But then I found out about Simple System, and I was so impressed with your concept (super-fast character creation, little to no mental math during encounters, no dice!) that I was instantly on board. It does pretty much exactly what I intended my 2d6 system to do, but much better. Well done!
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Post by Dashing Inventor on Jul 3, 2014 23:37:57 GMT -8
I share your love for game systems (obviously) and am very interested to see what your simulation comes up with. I was considering doing the same thing, but ended up just doing most of it manually (flipping lots and lots of cards) which is time consuming but very comforting when it confirms your math.
The reason I don't treat checks as positive and strikes as negative is because it would require more math than just adding them up respectively and tossing out the lesser result. Its all part of Simple System's design philosophy to speed things up - one way of doing that is by avoiding math. The most math you have to do when using Simple System is literally counting how many spaces your character can move.
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Ziphion
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Post by Ziphion on Jul 4, 2014 9:10:12 GMT -8
I like that, it means you just have to look down at the cards and see your result instead of having to interpret ✓✓✓✗✗ as ✓.
Haven't had a chance to work on the simulator since we've got family staying with us for the holiday, but I can work on it next week.
Happy 4th, by the way!
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Post by directedbyme on Jul 7, 2014 11:52:42 GMT -8
Great news! Love the new color scheme and probability adjustments.
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Ziphion
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Post by Ziphion on Jul 8, 2014 7:54:17 GMT -8
(re: kickstarter update to colors (b>g>y>r) and probabilities)
Awesome, thanks so much for making the adjustment! Before you make any big changes, I wanted to float another possibility by you regarding the deck probabilities. As it is, deck rotations only improve your odds by about 7.5% (haven't tested how much flipping doubles or triples improves your odds yet; sim is almost functional). That seems a bit low to me. I have a feeling (tests to follow) that spending a hero point/card increases your chances by much more than a rotation, or maybe even two rotations. It might be nice to slightly increase the difference between blue & green, green & yellow, and yellow & red so that players feel like their choices and character builds matter, and so that the mechanics better support their roleplay.
(Side note: the reason I worry about this is because after DMing two groups in 4e for a while, I noticed both groups becoming frustrated that, a little too often, the dunce paladin with +1 Arcana skill is able to succeed on an Arcana check that the wizard with +9 Arcana failed. This happened due to the wide variance of rolling a d20, so I fixed that with replacing all skill-based d20 rolls with 2d10. That tweaked the probability to make exceptional results a bit more rare and exciting.)
Here's my suggested change:
Still the same number of positive/negative results per color, but I've just traded more single-successes for triple-successes (and likewise for failures). I estimate this makes rotating the deck improve your odds by about 9%, but I won't know for sure until I mess with my simulation tonight after work and get some results.
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Post by directedbyme on Jul 8, 2014 9:17:55 GMT -8
I like the idea of a 9% increase. It makes the lowering/raising of the difficulty that much more significant.
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Ziphion
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Post by Ziphion on Jul 8, 2014 9:25:34 GMT -8
Some thoughts about pips!
Distribution in the demo deck (with colors changed as per the update):
| blue | green | yellow | red | | 1 | 21 | 18 | 7 | 3 | | 2 | 8 | 9 | 19 | 19 | | 3 | 1 | 3 | 4 | 8 |
avg value |
1.333 |
1.5 |
1.9 |
2.167 |
So green is on average 1/6 (0.167) pips higher than blue, yellow is 2/5 (0.4) pips higher than green, and red is 4/15 (0.267) pips higher than yellow. Here's a suggested distribution that smooths out these differences:
| blue | green | yellow | red | | 1 | 22 | 15 | 8 | 1 | | 2 | 7 | 9 | 11 | 13 | | 3 | 1 | 6 | 11 | 16 |
avg value |
1.3 |
1.7 |
2.1 |
2.5 |
This way the difference between each color is always 2/5 (0.4). Again, I'm not yet sure how double/triple flips will compare.
Thoughts?
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Ziphion
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Post by Ziphion on Jul 8, 2014 9:53:42 GMT -8
Actually, I can't remember... how do you handle pips on multiple flips? You can't just add them up, otherwise the average for 2x blue is just twice that of blue, and same for red.
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Post by directedbyme on Jul 8, 2014 10:29:26 GMT -8
I think you are supposed to use the last flip for the pips.
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Ziphion
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Post by Ziphion on Jul 8, 2014 10:37:53 GMT -8
Just use the last flip... so 2x or 3x blue is just the same average pip number as 1x blue? You know, come to think of it, I haven't seen a weapon or armor that had a rating beyond the four basic colors that would require multiple flips anyway. So it may be a non-issue.
Also - I just realized that in the most recent Kickstarter update, DI mentioned he'd be reevaluating pips anyway! So I'll just wait and see what the changes are before spending a lot of time thinking about 'em.
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Ziphion
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Post by Ziphion on Jul 8, 2014 16:18:21 GMT -8
Preliminary results! These tables indicate the probability that the color listed in the left column will win an opposed check against the color listed in the top row. The tests were performed using a 36-card deck simulated in C++, reshuffled whenever the deck ran out of cards; each value in each table represents one million head-to-head flips. I will test out different shuffling strategies later (shuffling after 3 criticals, etc). Before that, I'll implement multiple flips and make similar tables to the ones below.
Original demo deck: (colors altered as per the recent update)
| blue | green | yellow | red | | blue | 50.0% | 41.3% | 33.3% | 27.4% | | green | 58.7% | 50.0% | 41.3% | 34.2% | | yellow | 66.7% | 58.7% | 50.0% | 42.4% | | red | 72.6% | 65.8% | 57.6% | 50.0% |
Odds improve by, on average:
8.1% when rotating from blue to green.
8.4% when rotating from green to yellow.
7.0% when rotating from yellow to red.
7.8% (± 0.9%) overall.
Altered Deck 1:
| blue | green | yellow | red | | blue | 50.0% | 41.7% | 34.0% | 26.8% | | green | 58.3% | 50.0% | 41.9% | 34.0% | | yellow | 66.0% | 58.1% | 50.0% | 41.6% | | red | 73.2% | 66.0% | 58.3% | 50.0% |
Odds improve by, on average:
7.9% when rotating from blue to green.
7.9% when rotating from green to yellow.
7.9% when rotating from yellow to red.
7.9% (± 0.4%) overall.
Altered Deck 2: (my post from earlier today)
| blue | green | yellow | red | | blue | 50.0% | 40.4% | 31.5% | 23.3% | | green | 59.6% | 50.0% | 40.6% | 31.5% | | yellow | 68.4% | 59.4% | 50.0% | 40.4% | | red | 76.7% | 68.5% | 59.6% | 50.0% |
Odds improve by, on average:
9.1% when rotating from blue to green.
9.1% when rotating from green to yellow.
9.1% when rotating from yellow to red.
9.1% (± 0.5%) overall.
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Post by Arcanet on Jul 8, 2014 16:49:01 GMT -8
I like the 9% improvement, as directedbyme said, makes attributes and proper use of skills matter more. Bonus points for improving the usefulness of skillful tactics, as being a dunce and getting a Complication hurts more.  |
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